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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 Thermophoresis Effect On Unsteady Free Convection Heat And Mass Transfer In A Walters-B Fluid Past A Semi Infinite PlateC. Sudhakar*, N. Bhaskar Reddy*, #B. Vasu**, V. Ramachandra Prasad** *Department of Mathematics, Sri Venkateswara University, Tirupati, A.P, India** Department of Mathematics, Madanapalle Institute of Technology and Sciences, Madanapalle-517325, India.ABSTRACT The effect of thermophoresis particle gas flow, the thermophoretic deposition plays andeposition on unsteady free convective, heat and important role in a variety of applications such asmass transfer in a viscoelastic fluid along a semi- the production of ceramic powders in highinfinite vertical plate is investigated. The temperature aerosol flow reactors, the production ofWalters-B liquid model is employed to simulate optical fiber performs by the modified chemicalmedical creams and other rheological liquids vapor deposition (MCVD) process and in a polymerencountered in biotechnology and chemical separation. Thermophoresis is considered to beengineering. The dimensionless unsteady, important for particles of 10 m in radius andcoupled and non-linear partial differential temperature gradient of the order of 5 K/mm.conservation equations for the boundary layer Walker et al. [1] calculated the deposition efficiencyregime are solved by an efficient, accurate and of small particles due to thermophoresis in a laminarunconditionally stable finite difference scheme of tube flow. The effect of wall suction andthe Crank-Nicolson type. The behavior of thermophoresis on aerosol-particle deposition fromvelocity, temperature and concentration within a laminar boundary layer on a flat plate was studiedthe boundary layer has been studied for by Mills et al. [2]. Ye et al. [3] analyzed thevariations in the Prandtl number (Pr), thermophoretic effect of particle deposition on a freeviscoelasticity parameter (), Schmidt number standing semiconductor wafer in a clean room.(Sc), buoyancy ration parameter (N) and Thakurta et al. [4] computed numerically thethermophoretic parameter ( ). The local skin- deposition rate of small particles on the wall of afriction, Nusselt number and Sherwood number turbulent channel flow using the direct numericalare also presented and analyzed graphically. It is simulation (DNS). Clusters transport and depositionobserved that, an increase in the thermophoretic processes under the effects of thermophoresis wereparameter ( ) decelerates the velocity as well as investigated numerically in terms of thermal plasmaconcentration andaccelerates temperature. In deposition processes by Han and Yoshida [5]. Inaddition, the effect of the thermophoresis is also their analysis, they found that the thickness of thediscussed for the case of Newtonian fluid. concentration boundary layer was significantly suppressed by the thermophoretic force and it wasKey words: Finite difference method,semi-infinite concluded that the effect of thermophoresis plays avertical plate, Thermophoresis effect, Walters-B more dominant role than that of diffusion. Recently,fluid, unsteady flow. Alam et al. [6] investigated numerically the effect of thermophoresis on surface deposition flux on hydromagnetic free convective heat mass transfer1. INTRODUCTION flow along a semi- infinite permeable inclined flat Prediction of particle transport in non-isothermal gas flow is important in studying the plate considering heat generation. Their results show that thermophoresis increases surface mass fluxerosion process in combustors and heat exchangers, significantly. Recently, Postalnicu [7] has analyzedthe particle behavior in dust collectors and thefabrications of optical waveguide and the effect of thermophoresis particle deposition insemiconductor device and so on. Environmental free convection boundary layer from a horizontal flat plate embedded in porous medium.regulations on small particles have also become The study of heat and mass transfer in non-more stringent due to concerns about atmospheric Newtonian fluids is of great interest in manypollution. operations in the chemical and process engineering When a temperature gradient is established industries including coaxial mixers, bloodin gas, small particles suspended in the gas migrate oxygenators [8], milk processing [9], steady-statein the direction of decreasing temperature. The tubular reactors and capillary column inverse gasphenomenon, called thermophoresis, occurs becausegas molecules colliding on one side of a particle chromatography devices mixing mechanism bubble-have different average velocities from those on the drop formation processes [10] dissolution processesother side due to the temperature gradient. Hence and cloud transport phenomena. Many liquidswhen a cold wall is placed in the hot particle-laden possess complex shear-stress relationships which 2080 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095deviate significantly from the Newtonian (Navier- deformable stretching surface. They found that aStokes) model. External thermal convection flows in boundary layer is generated formed when thesuch fluids have been studied extensively using inviscid free-stream velocity exceeds the stretchingmathematical and numerical models and often velocity of the surface and the flow is acceleratedemploy boundary-layer theory. Many geometrical with increasing magnetic field. This study alsoconfigurations have been addressed including flat identified the presence of an inverted boundary layerplates, channels, cones, spheres, wedges, inclined when the surface stretching velocity exceeds theplanes and wavy surfaces. Non-Newtonian heat velocity of the free stream and showed that for thistransfer studies have included power-law fluid scenario the flow is decelerated with increasingmodels [11-13] i.e. shear-thinning and shear magnetic field. Rajagopal et al [32] obtained exactthickening fluids, simple viscoelastic fluids [14, 15], solutions for the combined nonsimilarCriminale-Ericksen-Fibley viscoelastic fluids [16], hydromagnetic flow, heat, and mass transferJohnson-Segalman rheological fluids [17], Bingham phenomena in a conducting viscoelastic Walters-Byield stress fluids [18], second grade (Reiner-Rivlin) fluid percolating a porous regime adjacent to aviscoselastic fluids [19] third grade viscoelastic stretching sheet with heat generation, viscousfluids [20], micropolar fluids [21] and bi-viscosity dissipation and wall mass flux effects, usingrheological fluids [22]. Viscoelastic properties can confluent hypergeometricfunctions for differentenhance or depress heat transfer rates, depending thermal boundary conditions at the wall.upon the kinematic characteristics of the flow field Steady free convection heat and massunder consideration and the direction of heat transfer flow of an incompressible viscous fluid pasttransfer. The Walters-B viscoelastic model [23] was an infinite or semi-infinite vertical plate is studieddeveloped to simulate viscous fluids possessing since long because of its technological importance.short memory elastic effects and can simulate Pohlhausen [33], Somers [34] and Mathers et al.accurately many complex polymeric, [35] were the first to study it for a flow past a semi-biotechnological and tribological fluids. The infinite vertical plate by different methods. But theWalters-B model has therefore been studied first systematic study of mass transfer effects on freeextensively in many flow problems. Soundalegkar convection flow past a semi-infinite vertical plateand Puri[24] presented one of the first mathematical was presented by Gebhart and pera [36] whoinvestigations for such a fluid considering the presented a similarity solution to this problem andoscillatory two-dimensional viscoelastic flow along introduced a parameter N which is a measure ofan infinite porous wall, showing that an increase in relative importance of chemical and thermalthe Walters elasticity parameter and the frequency diffusion causing a density difference that drives theparameter reduces the phase of the skin-friction. flow. Soundalgekar and Ganesan [37] studiedRoy and Chaudhury [25] investigated heat transfer transient free convective flow past a semi-infinitein Walters-B viscoelastic flow along a plane wall vertical flat plate with mass transfer by usingwith periodic suction using a perturbation method Crank–Nicolson finite difference method. In theirincluding viscous dissipation effects. Raptis and analysis they observed that, an increase in N leads toTakhar [26] studied flat plate thermal convection an increase in the velocity but a decrease in theboundary layer flow of a Walters-B fluid using temperature and concentration. Prasad et al. [38]numerical shooting quadrature. Chang et al [27] studied Radiation effects on MHD unsteady freeanalyzed the unsteady buoyancy-driven flow and convection flow with mass transfer past a verticalspecies diffusion in a Walters-B viscoelastic flow plate with variable surface temperature andalong a vertical plate with transpiration effects. concentration Owing to the significance of thisThey showed that the flow is accelerated with a rise problem in chemical and medical biotechnologicalin viscoelasticity parameter with both time and processing (e.g. medical cream manufacture).distances close to the plate surface and that Therefore the objective of the present paper is toincreasing Schmidt number suppresses both velocity investigate the effect of thermophoresis on anand concentration in time whereas increasing unsteady free convective heat and mass transferspecies Grashof number (buoyancy parameter) flow past a semi infinite vertical plate using theaccelerates flow through time. Hydrodynamic robust Walters-B viscoelastic rheologicctal materialstability studies of Walters-B viscoelastic fluids model.A Crank-Nicolson finite difference scheme iswere communicated by Sharma and Rana [28] for utilized to solve the unsteady dimensionless,the rotating porous media suspension regime and by transformed velocity, thermal and concentrationSharma et al [29] for Rayleigh-Taylor flow in a boundary layer equations in the vicinity of theporous medium. Chaudhary and Jain [30] studied vertical plate. The present problem has to thethe Hall current and cross-flow effects on free and author’ knowledge not appeared thus far in theforced Walters-B viscoelastic convection flow with literature. Another motivation of the study is to erthermal radiative flux effects. Mahapatra et al [31] observed high heat transfer performance commonlyexamined the steady two-dimensional stagnation- attributed to extensional investigate thestresses inpoint flow of a Walters-B fluid along a flat viscoelastic boundary layers [25] 2081 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 2. CONSTITUTIVE EQUATIONS FOR This rheological model is very versatile and robust THE WALTERS-B VISCOELASTIC and provides a relatively simple mathematical FLUID formulation which is easily incorporated into Walters [23] has developed a physically boundary layer theory for engineering applications accurate and mathematically amenable model for [25, 26]. the rheological equation of state of a viscoelastic fluid of short memory. This model has been shown 3. MATHEMATICAL MODEL: to capture the characteristics of actual viscoelastic An unsteady two-dimensional laminar free polymer solutions, hydrocarbons, paints and other convective flow of a viscoelastic fluid past a semi- chemical engineering fluids. The Walters-B model infinite vertical plate is considered. The x-axis is generates highly non-linear flow equations which taken along the plate in the upward direction and the are an order higher than the classical Navier-Stokes y-axis is taken normal to it. The physical model is (Newtonian) equations. It also incorporates elastic shown in Fig.1a. properties of the fluid which are important in extensional behavior of polymers. The constitute u equations for a Walters-B liquid in tensorial form v may be presented as follows: X pik pgik pik * (1) Boundary 1 Tw , Cw layer pik * 2 t t * e1ik t * dt * (2) T , C N t t* e t t * / d (3) o Y 0where pik is the stress tensor, p is arbitrary isotropic pressure,g ik is the metric tensor of a fixed coordinate system xi, eik 1 isthe rate of strain tensor and N is the distribution functionof relaxation times, . The following generalized form of (2) has Fig.1a. Flow configuration and coordinate systembeen shown by Walters [23] to be valid for all classes of motionand stress. Initially, it is assumed that the plate and the x 1pik x, t 2 t t *m *r e1mr x*t * dt * (4) * * fluid are at the same temperature T and x x concentration level C everywhere in the fluid. At time, t >0, Also, the temperature of the plate and in which xi xi * * x, t, t denotes the position at * the concentration level near the plate are raised to time t* of the element which is instantaneously at Tw and C w respectively and are maintained the position, xi, at time, t. Liquids obeying the constantly thereafter. It is assumed that the relations (1) and (4) are of the Walters-B’ type. For such fluids with short memory i.e. low relaxation concentration C of the diffusing species in the times, equation (4) may be simplified to: binary mixture is very less in comparison to the other chemical species, which are present, and hence the Soret and Dufour effects are negligible. It is also e1ik p*ik x, t 20e 2k0 (5) assumed that there is no chemical reaction between 1 ik t the diffusing species and the fluid. Then, under the above assumptions, the governing boundary layer in which 0 N d defines the limiting equations with Boussinesq’s approximation are 0 viscosity at low shear rates, u v Walters-B’ 0 (6) x y k0 N d is the Walters-B’ viscoelasticity 0 parameter and is the convected time derivative. t 2082 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095u u u k Tw T u v g T T g * C C t x y T0 2u 3u (7) Typical values of are 0.01, 0.05 and 0.1 2 k0 2 corresponding to approximate values of y y t k Tw T equal to 3.15 and 30 k for a reference T T T k 2T u v (8) temperature of T0 =300 k. t x y c p y 2 On introducing the following non-dimensional C C C 2C quantities u v D 2 cvt (9) t x y y y x yGr1/4 uLGr 1/2 X ,Y ,U , L L The initial and boundary conditions are T T C C tL2 1/2 (11) t 0 : u 0, v 0, T T , C C 0 T ,C , t Gr , Tw T Cw C t 0 : u 0, v 0, T Tw , C Cw at y 0 * (Cw C ) k Gr1/2 u 0, T T , C C at x 0 N , 0 2 , (Tw T ) L u 0, T T , C C as y (10) g (Tw T ) Gr , Where u, v are velocity components in x u0 3 and y directions respectively, t - the time, g – the kL Tw T Pr , Sc , acceleration due to gravity, - the volumetric D Tw coefficient of thermal expansion, - the * Equations (6), (7), (8), (9) and (10) are reduced to volumetric coefficient of expansion with the following non-dimensional form concentration, T -the temperature of the fluid in U U 0 (12) the boundary layer, C -the species concentration X Y in the boundary layer, Tw - the wall temperature, U U U 2U 3U (13) U V T NC 2 T - the free stream temperature far away from the t X Y Y 2 Y t plate, C w - the concentration at the plate, C - the free stream concentration in fluid far away from the plate, - the kinematic viscosity, - the thermal T T T 1 2T diffusivity, - the density of the fluid and D - the U V (14) species diffusion coefficient. t X Y Pr Y 2 In the equation (4), the thermophoretic velocity C C C 1 2C U V Vt was given by Talbot et al. [39] as t X Y Sc Y 2 T kv T C T 2T VT kv C 2 (15) Tw y 1 Tw Gr 4 Y Y Y Where Tw is some reference temperature, the value The corresponding initial and boundary conditions of kv represents the thermophoretic diffusivity, and are k is the thermophoretic coefficient which ranges in t 0 : U 0, V 0, T 0, C 0 value from 0.2 to 1.2 as indicated by Batchelor and t 0 : U 0, V 0, T 1, C 1 at Y 0, Shen [40] and is defined from the theory of Talbot et al [39] by U 0, V 0, C 0 as X 0 (16) 2Cs (g / p Ct K n ) 1 K n (C1 C2e Cs / Kn ) U 0, V 0, C 0 as Y k (1 3Cm K n ) 1 2g / p 2Ct K n Where Gr is the thermal Grashof number, Pr is the fluid Prandtl number, Sc is the Schmidt A thermophoretic parameter can be defined (see number, N is the buoyancy ratio parameter, is the Mills et al 2 and Tsai [41]) as follows; viscoelastic parameter and is the thermophoretic parameter. To obtain an estimate of flow dynamics at the barrier boundary, we also define several important 2083 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 rate functions at Y = 0. These are the dimensionless 1 wall shear stress function, i.e. local skin friction U in, 1 U in1j 1 U in, 1 U in1j 1 4x j 1, j 1, function, the local Nusselt number (dimensionless temperature gradient) and the local Sherwood U in, j 1 U in1, j 1 U in, j U in1, j (18) number (dimensionless species, i.e. contaminant transfer gradient) are computed with the following Vi ,nj1 Vi ,nj11 Vi ,nj Vi ,nj 1 mathematical expressions [48] 0 2 y 1 T XGr 4 U in, 1 U in, j U in, 1 U in1j U in, j U in1, j 3 U Y Y 0 X Gr 4 , Nu x j U n j 1, , t i, j 2 x Y Y 0 TY 0 1 C U in, 1 U in, 1 U in, j 1 U in, j 1 1 1 XGr 4 Vi ,nj j j Y Y 0 (17) 4y ShX CY 0 U in, 1 2U in, 1 U in, 1 U in, j 1 2U in, j U in, j 1 1 1 j j j We note that the dimensionless model 2 y (19) 2 defined by Equations (12) to (15) under conditions (16) reduces to Newtonian flow in the case of U in, 1 2U in, 1 U in, 1 U in, j 1 2U in, j U in, j 1 1 1 vanishing viscoelasticity i.e. when = 0 j j j 2 y t 2 4.NUMERICAL SOLUTION In order to solve these unsteady, non-linear 1 Ti ,nj1 Ti ,nj Cin, 1 Cin, j N 4 j coupled equations (12) to (15) under the conditions Gr 2 2 (16), an implicit finite difference scheme of Crank- Nicolson type has been employed. This method was originally developed for heat conduction problems Ti ,nj1 Ti ,nj Ti ,nj1 Ti 1,1j Ti ,nj Ti 1, j n n [42]. It has been extensively developed and remains U in, j one of the most reliable procedures for solving t 2 x partial differential equation systems. It is unconditionally stable. It utilizes a central Ti ,nj11 Ti ,nj11 Ti ,nj 1 Ti ,nj 1 (20) differencing procedure for space and is an implicit V n 4y i, j method. The partial differential terms are converted n 1 n 1 n 1 1 Ti , j 1 2Ti , j Ti , j 1 Ti , j 1 2Ti , j Ti , j 1 to difference equations and the resulting algebraic n n n problem is solved using a triadiagonal matrix 2 y 2 algorithm. For transient problems a trapezoidal rule Pr is utilized and provides second-order convergence. The Crank-Nicolson Method (CNM) scheme has Cin, 1 Cin, j Cin, 1 Cin1j Cin, j Cin1, j been applied to a rich spectrum of complex j U n j 1, t 2 x multiphysical flows. Kafousias and Daskalakis [43] i, j have employed the CNM to analyze the hydromagnetic natural convection Stokes flow for Cin, 1 Cin, 1 Cin, j 1 Cin, j 1 1 1 n j j air and water. Edirisinghe [44] has studied V 4y i, j efficiently the heat transfer in solidification of n 1 n 1 n 1 1 Ci , j 1 2Ci , j Ci , j 1 Ci , j 1 2Ci , j Ci , j 1 ceramic-polymer injection moulds with CNFDM. n n n Sayed-Ahmed [45] has analyzed the laminar 2 y 2 dissipative non-Newtonian heat transfer in the Sc entrance region of a square duct using CNDFM. Nassab [46] has obtained CNFDM solutions for the Cin, 1 Cin, 1 Cin, j 1 Cin, j 1 j 1 j 1 (21) unsteady gas convection flow in a porous medium 1 with thermal radiation effects using the Schuster- Gr 4 4y Schwartzchild two-flux model. Prasad et al [47] Ti , j 1 Ti , j 1 Ti , j 1 Ti , j 1 n 1 n 1 n n studied the combined transient heat and mass transfer from a vertical plate with thermal radiation 4y effects using the CNM method. The CNM method n 1 n 1 n 1 works well with boundary-layer flows. The finite n Ti , j 1 2Ti , j Ti , j 1 Ti ,nj 1 2Ti ,nj Ti ,nj 1 difference equations corresponding to equations (12) Ci , j 2 y 1 2 to (15) are discretized using CNM as follows Gr 4 2084 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095Here the region of integration is considered as a 5.RESULTS AND DISCUSSIONrectangle with X max 1 , Ymax 14 and where In order to get a physical insight into the problem, a parametric study is carried out toYmax corresponds to Ywhich lies well outside illustrate the effect of various governingboth the momentum and thermal boundary layers. thermophysical parameters on the velocity,The maximum of Y was chosen as 14, after some temparature, concentration, skin-friction, Nusseltpreliminary numerical experiments such that the last number and Sherwood number are shown in figurestwo boundary conditions of (19) were satisfied and tables. 5 In figures 2(a) to 2(c) we have presentedwithin the tolerance limit 10 . The mesh sizes havebeen fixed the variation of velocity U, temparature T andas XYwith concentration C versus (Y) with collective effects of timestep t0.01. The computations are executed thermophoretic parameter () at X = 0 for opposinginitially by reducing the spatial mesh sizes by 50% flow (N<0). In case of Newtonian fluids ( = 0), anin one direction, and later in both directions by 50%. increase in from 0.0 through 0.5 to maximumThe results are compared. It is observed that, in all value of 1.0 as depicted in figure 2(a) for opposingthe cases, the results differ only in the fifth decimal flow (N < 0) . Clearly enhances the velocity Uplace. Hence, the choice of the mesh sizes is verified which ascends sharply and peaks in close vicinity toas extremely efficient. The coefficients of the plate surface (Y=0). With increasing distance from the plate wall however the velocity U isUik, j andVi,kj ,appearing in the finite difference adversely affected by increasing thermophoreticequations are treated as constant at any one-time effect i.e. the flow is decelerated. Therefore close tostep. Here i designates the grid point along the X- the plate surface the flow velocity is maximized fordirection, j along the Y-direction and k in the time the case ofBut this trend is reversed asvariable, t. The values of U, V, T and C are known at we progress further into the boundary layer regime.all grid points when t = 0 from the initial conditions. The switchover in behavior corresponds toThe computations for U, V, T and C at a time level approximately Y =3.5, with increasing velocity(k + 1), using the values at previous time level k are profiles decay smoothly to zero in the free stream atcarried out as follows. The finite-difference equation the edge of the boundary layer. The opposite effect(21) at every internal nodal point on a particular is caused by an increase in time.A rise in fromilevel constitutes a tri diagonal system of - 6.36, 7.73 to 10.00 causes a decrease in flowequations and is solved by Thomas algorithm as velocity U near the wall in this case the maximumdiscussed in Carnahan et al. [45]. Thus, the values velocity arises for the least time progressed.Withof C are known at every nodal point at a particular i more passage of time t = 10.00 the flow isat k 1 time level. Similarly, the values of U th decelerated.Again there is a reverse in the response at Y =3.5, and thereafter velocity is maximized withand T are calculated from equations (19), (20) the greatest value of time. A similar response isrespectively, and finally the values of V are observed for the non-Newtonian fluid ( 0 ), butcalculated explicitly by using equation (18) at every clearly enhances the velocity very sharply and peaksnodal point on a particular i level at k 1 th highly in close vicinity to the plate surface compared in case of Newtonian fluid.time level. In a similar manner, computations are In figure 2(b), in case of Newtonian fluidscarried out by moving along i -direction. After (=0) and non-Newtonian fluids ( 0 ), thecomputing values corresponding to each i at a time thermophoretic parameter is seen to increaselevel, the values at the next time level are temperature throughout the bounder layer.Alldetermined in a similar manner. Computations are profiles increases from the maximum at the wall torepeated until steady state is reached. The steady zero in the free stream. The graphs show thereforestate solution is assumed to have been reached when that increasing thermophoretic parameter heatedthe absolute difference between the values of the the flow. With progression of time, however thevelocity U, temperature T, as well as concentration 5 temperature T is consistently enhanced i.e. the fluidC at two consecutive time steps are less than 10 at is cool as time progress.all grid points. The scheme is unconditionally stable.The local truncation error isO(t 2 X 2 Y 2 ) and it tends to zero as t,X, and Ytend to zero. It follows that theCNM scheme is compatible. Stability andcompatibility ensure the convergence. 2085 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 In figure 2(c) theopposite response isobserved for the concentration field C. In case of The switchover in behavior corresponds toNewtonian fluids (=0) and non-Newtonian fluids ( approximately Y =3.5, with increasing velocity 0 ), the thermophoretic parameter profiles decay smoothly to zero in the free stream atincreases, the concentration throughout the the edge of the boundary layer.The opposite effect isboundary layer regime (0<Y<14) decreased. caused by an increase in time. A rise in time t from 6.36, 7.73 to 10.00 causes a decrease in flow In figures 3(a) to 3(c) we have presentedthe variation of velocity U, temparature T andconcentration C versus (Y) with collective effects ofthermophoretic parameter at X = 0 for aiding velocity U near the wall in this case theflow. In case of Newtonian fluids ( = 0), an maximum velocity arises for the least timeincrease in from 0.0 through 0.5 to maximum progressed.With more passage of time t = 10.00 thevalue of 1.0 as depicted in figure 3(a) for aiding flow is decelerated. Again there is a reverse in theflow (N>0). Clearly enhances the velocity U which response at Y =3.5, and thereafter velocity isascends sharply and peaks in close vicinity to the maximized with the greatest value of time. A similarplate surface (Y=0). With increasing distance from response is observed for the non-Newtonian fluid (the plate wall however the velocity U is adversely 0 ), but clearly enhances the velocity veryaffected by increasing thermophoretic effect i.e. sharply and peaks highly in close vicinity to thetheflow is decelerated. Therefore close to the plate plate surface compared in case of Newtonian fluid.surface the flow velocity is maximized for the caseof But this trend is reversed as we progressfurther into the boundary layer regime. 2086 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 depicted in figure 4(a), clearly enhances the velocity U which ascends sharply and peaks in close vicinity to the plate surface (Y=0),with increasing distance from the plate wall the velocity U is adversely affected by increasing viscoelasticity i.e. the flow is decelerated. Therefore close to the plate surface the flow velocity is maximized for the case ofnon- Newtonian fluid( 0 ). The switchover in behavior corresponds to approximately Y=2. In figure 3(b), in case of Newtonian fluids(=0) and non-Newtonian fluids ( 0 ), thethermophoretic parameter is seen to decreasetemperature throughout the bounder layer. Allprofiles decreases from the maximum at the wall tozero in the free stream. The graphs show thereforethat increasing thermophoretic parameter cools the With increasing Y, velocity profiles decayflow. With increasing of time t, the temperature T is smoothly to zero in the free stream at the edge ofconsistently enhanced i.e. the fluid is heated as time theboundary layer. Pr<1 physically corresponds toprogress. cases where heat diffuses faster than momentum. In In figure 3(c) a similar response is the case of water based solvents i.e. Pr = 7.0, aobserved for the concentration field C. In case of similar response is observed for the velocity field inNewtonian fluids (=0) and non-Newtonian fluids ( figure 4(a). In figure 4(b), in case of air based solvents 0 ), the thermophoretic parameter increases, i.e. Pr = 0.71, an increase in viscoelasticity the concentration throughout the boundary layer increasing from 0.000, 0.003 to 0.005, temperatureregime (0<Y<14) decreased. All profiles decreases T is markedly reduced throughout the boundaryfrom the maximum at the wall to zero in the free layer. In case of water based solvents i.e. Pr = 7.0stream. also a similar response is observed, but it is very Figures 4(a) to 4(c) illustrate the effect of closed to the plate surface. The descent isPrandtl number (Pr), Viscoelastic parameter () and increasingly sharper near the plate surface for highertime t on velocity (U), temperature (T) and Pr values a more gradual monotonic decay isconcentration (C) without thermophoretic effect witnessed smaller Pr values in this case, cause a(=0) at X=1.0.Pr defines the ratio of momentum thinner thermal boundary layer thickness and morediffusivity () to thermal diffusivity. In case of air uniform temperature distributions across thebased solvents i.ePr = 0.71, an increase in from boundary layer. Smaller Pr fluids possess higher0.000, 0.003 and the maximum value of 0.005 as thermal conductivities so that heat can diffuse away 2087 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095from the plate surface faster than for higher Pr fluids T is markedly reduced throughout the boundary(thicker boundary layers). Our computations show layer. In case of water based solvents i.e. Pr = 7.0that a rise in Pr depresses the temperature function, also a similar response is observed, but it is verya result constant with numerous other studies on closed to the plate surface. A similar response iscoupled heat and mass transfer. For the case of Pr = observed for the concentration field C in figure1, thermal and velocity boundary layer thickness are 5(c).In both cases Pr = 0.71 and Pr = 7.0, when equal. increasing from 0.000, 0.003 to 0.005, concentration A similar response is observed for the C also reduced throughout the boundary layerconcentration field C in figure 4(c). In both cases Pr regime (0<Y<14). All profiles decreases from the= 0.71 and Pr = 7.0, when increasing from 0.000, maximum at the wall to zero in the free stream.0.003 to 0.005, concentration C also reducedthroughout the boundary layer regime (0<Y<14).All profiles decreases from the maximum at the wallto zero in the free stream. Figures 6(a) to 6(c) depict the distributions of velocity U, temperature T and concentration C versus coordinate (Y) for various Schmidtnumbers (Sc) with collective effects of thermophoretic parameter () in case of Newtonian fluids (=0) and time (t), close to the leading edge at X = 1.0, are shown. Correspond to Schmidt number Sc=0.6 an increase in from 0.0 through 0.5 to 1.0 as depicted in figure 6(a), clearly enhances the velocity U which ascends sharply and peaks in close vicinity to the plate surface (Y=0). With increasing distance from the plate wall however the velocity U is adversely affected by increasing thermophoretic effect i.e. the flow is decelerated. Figures 5(a) to 5(c) illustrate the effect ofPrandtl number (Pr), Viscoelastic parameter () andtime t on velocity U, temperature T andconcentration C with thermophoretic effect ( =0.5) at X=1.0. In case of air based solvents i.ePr =0.71, an increase in from 0.000, 0.003 and themaximum value of 0.005 as depicted in figure 5(a),clearly enhances the velocity U which ascendssharply and peaks in close vicinity to the platesurface (Y=0). With increasing Y, velocity profilesdecay smoothly to zero in the free stream at the edge Therefore close to the plate surface theof the boundary layer. flow velocity is maximized for the case of In figure 5(b), in case of air based solvents But this trend is reversed as we progressi.e. Pr = 0.71, an increase in viscoelasticity further into the boundary layer regime. Theincreasing from 0.000, 0.003 to 0.005, temperature switchover in behavior corresponds to 2088 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095approximately Y =3.5, with increasing velocityprofiles decay smoothly to zero in the free stream atthe edge of the boundary layer.A similar response isobserved in case of Schmidt number Sc=2.0 also. maximum concentrations will be associated with this case (Sc = 0.6).For Sc > 1, momentum will diffuse faster than species causing progressively lower concentration values. With a increase in molecular diffusivity concentration boundary layer With higher Sc values the gradient of thickness is therefore increased. For the special casevelocity profiles is lesser prior to the peak velocity of Sc = 1, the species diffuses at the same rate asbut greater after the peak. momentum in the viscoelastic fluid. Both In figure 6(b), in case of Newtonian fluids concentration and boundary layer thicknesses are(=0) and for Schmidt number Sc=0.6, and 2.0 the the same for this case. An increase in Schmidtincreasing in thermophoretic parameter is seen to number effectively depresses concentration valuesdecrease the temperature throughout the bounder in the boundary layer regime since higher Sc valueslayer. All profiles decreases from the maximum at will physically manifest in a decrease of molecularthe wall to zero in the free stream. The graphs show diffusivity (D) of the viscoelastic fluid i.e. atherefore that decreasing thermophoretic parameter reduction in the rate of mass diffusion. Lower Sccools the flow. With progression of time, however values will exert the reverse influence since theythe temperature T is consistently enhanced i.e. the correspond to higher molecular diffusivities.fluid is heated as time progress. Concentration boundary layer thickness is therefore In figure 6(c) a similar response is considerably greater for Sc = 0.6 than for Sc = 2.0.observed for the concentration field C. In case of Figures 7(a) to 7(c) depict the distributions ofNewtonian fluids (=0) and for Schmidt number velocity U, temperature T and concentration CSc=0.6, and 2.0, the increasing in thermophoretic versus coordinate (Y) for various Schmidt numbersparameter increases the concentration (Sc) with collective effects of thermophoreticthroughout the boundary layer regime (0<Y<14). parameter () in case of non-Newtonian fluids(All profiles increases from the maximum at the wallto zero in the free stream. Sc defines the ratio of 0 ) and time (t), close to the leading edge at Xmomentum diffusivity () to molecular diffusivity = 1.0, are shown. Correspond to Schmidt number(D). For Sc<1, species will diffuse much faster than Sc=0.6 anmomentum so that 2089 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 In figure 7(c) a similar response is observed for the concentration field C. In case of non-Newtonian fluids ( 0 ) and for Schmidt number Sc=0.6, and 2.0, the increasing in thermophoretic parameter increases the concentration throughout the boundary layer regime (0<Y<14). All profiles increases from the maximum at the wall to zero in the free stream. Sc defines the ratio of momentum diffusivity (n) to molecular diffusivity (D). For Sc<1, species will diffuse much faster than momentum so that maximum concentrations will be associated with this case (Sc = 0.6). For Sc > 1, momentum will diffuse faster than species causing progressively lower concentration values. With a increase in molecular diffusivity concentration boundary layer thickness is therefore increased. For the special case of Sc = 1, the species diffuses at the same rate as momentum in the viscoelastic fluid. Both concentration and boundary layer thicknesses are the same for this case. An increase in Schmidt number effectivelydepresses concentration values in the boundary layer regime since higher Sc values will physically manifest in a decrease of molecular diffusivity (D) of the viscoelastic fluid i.e. a reduction in the rate of mass diffusion. Lower Sc values will exert the reverse influence since they correspond to higher molecular diffusivities. Concentration boundary layer thickness is therefore considerably greater for Sc = 0.6 than for Sc = 2.0. Figures 8a to 8c present the effects of buoyancy increase in from 0.0 through 0.5 to 1.0 ratio parameter, N on U, T and C profiles. Theas depicted in figure 7(a),clearly enhances the maximum time elapse to the steady state scenariovelocity U which ascends sharply and peaks in close accompanies the only negative value of N i.e. N = -vicinity to the plate surface (Y=0). With increasing 0.5. For N = 0 and then increasingly positive valuesdistance from the plate wall however the velocity U of N up to 5.0, the time taken, t, is steadily reduced.is adversely affected by increasing thermophoretic As such the presence of aidingbuoyancy forceseffect i.e. the flow is decelerated. Therefore close to (both thermal and species buoyancy force acting inthe plate surface the flow velocity is maximized for unison) serves to stabilize the transient flow regime.the case of But this trend is reversed as we * Cw C progress further into the boundary layer regime. The The parameter N and expressesswitchover in behavior corresponds to Tw T approximately Y =3.5, with increasing velocity the ratio of the species (mass diffusion) buoyancyprofiles decay smoothly to zero in the free stream at force to the thermal (heat diffusion) buoyancy force.the edge of the boundary layer. A similar response is When N = 0 the species buoyancy term, NCobserved in case of Schmidt number Sc=2.0 also. vanishes and the momentum boundary layerAll profiles descend smoothly to zero in the free equation (13) is de-coupled from the speciesstream. With higher Sc values the gradient of diffusion (concentration) boundary layer equationvelocity profiles is lesser prior to the peak velocity (15). Thermal buoyancy does not vanish in thebut greater after the peak. momentum equation (13) since the term T is not In figure 7(b), in case of non-Newtonian affected by the buoyancy ratio.fluids ( 0 ) and for Schmidt number Sc=0.6, and2.0 the increasing in thermophoretic parameter isseen to decrease the temperature throughout thebounder layer. All profiles decreases from themaximum at the wall to zero in the free stream. Thegraphs show therefore that decreasingthermophoretic parameter cools the flow. Withprogression of time, however the temperature T isconsistently enhanced i.e. the fluid is heated as timeprogress. 2090 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 In figures 9a to 9c the variation of dimensionless local skin friction (surface shear stress), X Nusselt , number (surface heat transfer gradient), Nu X and the Sherwood number (surface concentration gradient), ShX , versus axial coordinate (X) for various viscoelasticity parameters () and time tare illustrated.Shear stress is clearly enhanced with increasing viscoelasticity (i.e. stronger elastic effects) i.e. the flow is accelerated, a trend consistent with our earlier computations in figure 9a. The ascent in shear stress is very rapid from the leading edge (X = 0) but more gradual as we progress along the plate surface away from the plane.With an increase in time, t, shear stress, X is When N < 0 we have the case of opposingbuoyancy. An increase in N from -0.5, through 0, 1, increased.Increasing viscoelasticity () is observed2, 3, to 5 clearly accelerates the flow i.e. induces a in figure 9b to enhance local Nusselt number, Nu Xstrong escalation in stream wise velocity, U, close to values whereas they areagain increased with greaterthe wall; thereafter velocities decay to zero in the time.Similarly in figure 9c the local Sherwoodfree stream. At some distance from the plate surface,approximately Y = 2.0, there is a cross-over in number ShXprofiles. Prior to this location the above trends areapparent. However after this point, increasinglypositive N values in fact decelerate the flow.Therefore further from the plate surface, negative Ni.e. opposing buoyancy is beneficial to the flowregime whereas closer to the plate surface it has aretarding effect. A much more consistent responseto a change in the N parameter is observed in figure8b, where with a rise from -0.5 through 0, 1.0, 2.0,3.0 to 5.0 (very strong aiding buoyancy case) thetemperature throughout the boundary layer isstrongly reduced. As with the velocity field (figure8a), the time required to attain the steady statedecreases substantially with a positive increase in N.Aiding (assisting) buoyancy therefore stabilizes thetemperature distribution.A similar response isevident for the concentration distribution C, Whichsshown in figure 8c, also decreases withpositiveincrease in N but reaches the steady stateprogressively faster. 2091 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 increase in time (t) both Shear stress and local Nusselt number are enhanced.values are elevated with an increase in elastic effectsi.e. a rise in from 0 (Newtonian flow) though r0.001, 0.003, 0.005 to 0.007 but depressed slightlywith time.Finally in figures 10a to 10c the influence ofThermophoretic parameter and time (t) on X With increasing values, local Sherwood number, ,Nu X and ShX , versus axial coordinate (X) are ShX , as shown in figure 10c, is boosteddepicted. considerably along the plate surface;gradients of the profiles are also found to diverge with increasing X values. However an increase in time, t, serves to increase local Sherwood numbers. 6. CONCLUSIONS A two-dimensional, unsteady laminar incompressible boundary layer model has been presented for the external flow, heat and mass transfer in a viscoelastic buoyancy-driven flow past a semi-infinite vertical plate under the influence of thermophoresis. The Walters-B viscoelastic model has been employed which is valid for short memory polymeric fluids. The dimensionless conservation equations have been solved with the well-tested, robust, highly efficient, implicit Crank Nicolson finite difference numerical method. The present computations have shown that increasingAn increase in from 0.0 through 0.3, 0.5, 1.0 to viscoelasticity accelerates the velocity and enhances1.5, strongly increases both X and Nu X along the shear stress (local skin friction), local Nusseltentire plate surface i.e. for all X. However with an number and local Sherwood number, but reduces 2092 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095temperature and concentration in the boundary aerosol- particle deposition from a laminarlayer. boundary layer on a flat plane, International journal of Heat and Mass7. NOMENCLATURE Transfer, 27, 1984, 1110-1113.x, y coordinates along the plate generator and [3] Y. Ye, D. Y. H. Pui, B.Y.H.Lia. S.Opiolka, normal to the generator respectively S. Opiolka, S.Blumhorst, H. Fisson,u, v velocity components along the x- and y- Thermophoretic effect of particle directions respectively deposition on a free standingg gravitational acceleration semiconductor wafer in a clean room,t time journal of Aerosol science,22(1), 1991, 63-t dimensionless time 72.Gr thermal Grashof number [4] D.G.Thakurta, M. Chen, J.B.Mclaughlim,k0 Walters-B viscoelasticity parameter K.Kontomaris, Tthermophoretic deposition of small particles in a direct numericalL reference length simulation of turbulent channel flow,Nu X Non-dimensional local Nusselt number Internationaljournal of Heat and massPr Prandtl number transfer, 41, 1998, 4167-4182.T temperature [5] P. Han, T. Yoshida, NumericalT dimensionless temperature investigationof thermophoretic effects ofC concentration cluster transport during thermal plasmaC dimensionless concentration deposition process,Journal of AppliedD mass diffusion coefficient Physics,91(4), 2002, 1814-1818.N Buoyancy ratio number [6] M.S.Alam, M.M. Rahman, M.A.Sattar,U, V dimensionless velocity components along Similarity solutions for hydromagnetic free the X- and Y- directions respectively convective heat and mass transfer flowX, Y dimensionless spatial coordinates along along a semi-infinite permeable inclined the plate generator and normal to the flat plate with heat generation and generator respectively thermophoresis. Nonlinear Anal modelSc Schmidt number control ,12(4), 2007, 433-445. [7] A Postelnicu, Effects of thermophoresisVt thermophoretic velocity particle deposition in free convectionShX non-dimensional local Sherwood number boundary layer from a horizontal flat plate embedded in a porousGreek symbols medium,Internationaljournal of Heat and thermal diffusivity mass transfer,50, 2007, 2981-2985. volumetric thermal expansion coefficient [8] A. R. Goerke, J. Leung and S. R. Wickramasinghe, Mass and momentum * volumetric concentration expansion transfer in blood oxygenators, Chemical coefficient Engineering Science, 57(11), 2002, 2035- viscoelastic parameter 2046. thermophoretic parameter [9] W. Wang and G. Chen, Heat and mass kinematic viscosity transfer model of dielectric-material-t dimensionless time-step assisted microwave freeze-drying of skimX dimensionless finite difference grid size in milk with hygroscopic effect, Chemical X-direction Engineering Science, 60(23), 2005, 6542-Y dimensionless finite difference grid size in 6550. Y-direction [10] A. B. Jarzebski and J. J. Malinowski, x dimensionless local skin-friction Transient mass and heat transfer from drops or bubbles in slow non-NewtonianSubscripts flows, Chemical Engineering Science,w condition on the wall 41(10), 1986,. 2575-2578.∞ free stream condition [11] R. PrakashBharti, R.P. Chhabra and V. Eswaran, Effect of blockage on heatREFERENCES transfer from a cylinder to power law [1] K.L. Walker, G. M. Homsy, F. T. Geying, liquids, Chemical Engineering Science, thermophoretic deposition of small 62(17), 2007, 4729-4741. particles in laminar tube,Journal of colloid [12] R. Mahalingam, S. F. Chan and J. M. and interface science,69, 1979, 138-147. Coulson, Laminar pseudoplastic flow heat [2] A. F. Mills, X. Hang, F. Ayazi, The effect transfer with prescribed wall heat flux, of wall suction and thermophoresis on 2093 | P a g e
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C. Sudhakar, N. Bhaskar Reddy, B. Vasu, V. Ramachandra Prasad / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2080-2095 Chemical Engineering Journal, 9(2), 1975, behaviour at small rates of shear is 161-166. characterized by a general linear equation[13] C.Kleinstreuer and T-Y. Wang, Mixed of state, Quarterly Journal of Mechanics convection heat and surface mass transfer and Applied. Mathematics, 15, 1962, 63- between power-law fluids and rotating 76. permeable bodies, Chemical Engineering [24] V. M. Soundalgekar and P. Puri On Science, 44(12), 1989, 2987-2994. fluctuating flow of an elastico-viscous fluid[14] J. L. White and A. B. Metzner, past an infinite plate with variable suction, Thermodynamic and heat transport Journal of Fluid Mechanics, 35(3), 1969, considerations for viscoelastic fluids, 561-573. Chemical Engineering Science, 20(12), [25] J. S. Roy and N. K. Chaudhury, Heat 1965, 1055-1062. transfer by laminar flow of an elastico-[15] A. V. Shenoy and R. A. Mashelkar, viscous liquid along a plane wall with Laminar natural convection heat transfer to periodic suction, Czechoslovak Journal of a viscoelastic fluid, Chemical Engineering Physics, 30(11) 1980, 1199-1209. Science, 33(6), 1978, 769-776. [26] A. A. Raptis and H. S. Takhar, Heat[16] S. Syrjälä, Laminar flow of viscoelastic transfer from flow of an elastico-viscous fluids in rectangular ducts with heat fluid, International Communication in Heat transfer: A finite element analysis, and Mass Transfer, 16(2), 1989, 193-197. International Communication in Heat and [27] T.B. Chang, Bég, O. Anwar, J. Zueco and Mass Transfer, 25(2), 1998, 191-204. M. Narahari, Numerical study of transient[17] T. Hayat, Z. Iqbal, M. Sajid, K. Vajravelu free convective mass transfer in a Walters- Heat transfer in pipe flow of a Johnson– B viscoelastic flow with wall suction, Segalman fluid, International Journal of Theoretical Applied Mechanics, Communication in Heat and Mass under review, 2009. Transfer, 35(10), 2008, 1297-1301. [28] V.Sharma and G.C.Rana, Thermosolutal[18] K. J. Hammad and G. C. Vradis, Viscous instability of Walters (model B) visco- dissipation and heat transfer in pulsatile elastic rotating fluid permeated with flows of a yield-stress fluid, International suspended particles and variable gravity Communication in Heat and Mass field in porous medium, International Transfer, 23(5), 1996, . 599-612. journal of Applied Mechanics Engineering,[19] O. A. Bég, Takhar H.S., Kumari M. and 6(4), 2001, 843-860. Nath G., Computational fluid dynamics [29] R. C. Sharma, P. Kumar and S. Sharma., modelling of buoyancy induced Rayleigh-Taylor instability of Walters- B viscoelastic flow in a porous medium with elastico-viscous fluid through porous magnetic field, Internationaljournal of medium, International journal of Applied Applied Mechanics Engineering, 6(1), Mechanics Engineering, 7(2), 2002, 433- (2001), 187-210. 444.[20] R. Bhargava, Rawat S., Takhar. H.S., Bég. [30] R. C. Chaudhary, and P. Jain, Hall effect O.A and V.R. Prasad, Numerical Study of on MHD mixed convection flow of a heat transfer of a third grade viscoelastic viscoelastic fluid past an infinite vertical fluid in non-Darcy porous media with porous plate with mass transfer and thermophysical effects, PhysicaScripta: radiation, Theoretical and Applied Proc. Royal Swedish Academy of Sciences, Mechanics, 33(4), 2006, 281-309. 77, 2008, 1-11. [31] T. Ray Mahapatra, S. Dholey and A. S.[21] O. A. Bég, R. Bhargava, K. Halim and H. Gupta, Momentum and heat transfer in the S. Takhar, Computational modeling of magnetohydrodynamic stagnation-point biomagnetic micropolar blood flow and flow of a viscoelastic fluid toward a heat transfer in a two-dimensional non- stretching surface, Meccanica, 42(3), 2007, Darcian porous medium, Meccanica, 43, 263-272. 2008, 391-410. [32] K. Rajagopal, P. H. Veena and V. K.[22] J. Zueco and O. A. Bég, Network Pravin, Nonsimilar Solutions for heat and numerical simulation applied to pulsatile mass transfer flow in an electrically non-Newtonian flow through a channel conducting viscoelastic fluid over a with couple stress and wall mass flux stretching sheet saturated in a porous effects, Internationaljournal of Applied medium with suction/blowing, Journal of Mathematics and Mechanics, 5(2), 2009, 1- Porous Media, 11(2), 2008, 219-230. 16. [33] E.Pohlhausen, Der[23] K. Walters, Non-Newtonian effects in Warmeaustrauschzwischenfestenkorpen some elastico-viscous liquids whose and FlussigkeitenmitKleinerReibung under 2094 | P a g e
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